Composite iteration for isogeometric collocation method using LSPIA and Schulz iteration

Abstract

The isogeometric least-squares collocation method (IGA-L) is an effective numerical technique for solving partial differential equations (PDEs), which utilizes the non-uniform rational B-splines (NURBS) to represent the numerical solution and constructs a system of equations using more collocation points than the number of unknowns. However, on the one hand, the convergence rate of the isogeometric collocation method is lower than that of the isogeometric Galerkin (IGA-G) method; on the other hand, the freedom of the numerical solution cannot be determined in advance to reach specified precision. In this paper, we model the solution of PDEs using IGA-L as a data fitting problem, in which the linear combination of the numerical solution and its derivatives is employed to fit the load function. Moreover, we develop a composite iterative method combining the least-squares progressive-iterative approximation (LSPIA) with the three-order Schulz iteration to solve the data fitting problem. The convergence of composite iterative method is proved, and the error bound is analyzed. Numerical results demonstrate that the convergence rate of the composite iterative method for IGA-L is nearly the same as that of IGA-G. Finally, we propose an incremental fitting algorithm with the composite iterative method, by which the freedom of numerical solution can be determined automatically to reach the specified fitting precision.